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Like in the title, if two mechanical waves interfere, for example sound waves with different frequencies, what is the frequency of the new wave? I know that for the beats, the frequency of the new sound we hear is equal to $\frac{\nu_1+\nu_2}{2}$, so if the beat is given by two waves with $\nu_1=440Hz$ and $\nu_2=441Hz$ the "new" wave has frequency $\nu=440,5Hz$ so that the sound is like an A and the "volume" changes with frequency $1Hz$. Is this relation true in general?

If we have a cosine wave with $\nu_1=200Hz$ and another cosine wave with $\nu_2=400Hz$ after the interfere do we hear a note corresponding to $\nu=300Hz$?

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    $\begingroup$ If two waves (mechanical, EM, whatever) in a linear medium, there is no new wave produced. If you are interested in the beat frequencies (which aren't a new wave), then it sounds like you understand them already. About what tones we hear when several tones reach our ear at the same time, that has a lot to do with psychoacoustics rather than physics. $\endgroup$
    – The Photon
    Aug 5, 2022 at 20:43
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    $\begingroup$ Another interesting set of effects to look into are acoustic sum and difference frequency generation. These actually emerge not as beats but as new features on the Fourier transform of the sound wave. $\endgroup$
    – Rishi
    Aug 5, 2022 at 20:48
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    $\begingroup$ You don't necessarily hear what is there. Using what you hear to determine the frequency spectrum of what is present doesn't work. It's like using your eyes and colour to try and determine the wavelength of light. Red + Blue = Purple, but $600nm + 495nm \ne 470nm$ $\endgroup$
    – DKNguyen
    Aug 5, 2022 at 21:01
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    $\begingroup$ In the online graphing calculator Desmos you can put two waves with general phases and frequencies and see how they add up, the resulting wave is in general very complicated looking and one really cannot define a frequency for it. $\endgroup$
    – Triatticus
    Aug 5, 2022 at 22:42
  • $\begingroup$ @ThePhoton Why there is no new wave produced? When we look at beats, don't we see a new wave (with that particular shape)? Isn't it the result of interfere of two waves with similar frequencies? $\endgroup$
    – Salmon
    Aug 6, 2022 at 0:45

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The middle ear can react nonlinearly, so instead of beats you can hear combination tones. There is a nice demonstration on the web page article Combination tones: Demonstrating the nonlinearity of the human ear

There are a lot of examples in photonics, and radio wave engineering, but the principle of nonlinear mixing also holds for surface acoustic wave in crystals ultrasound and other mechanical waves.

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